\section{Tiered pricing market}

\subsection{Market Model}
Tiered pricing might be more practical than flat and volume. According
to the report \cite{cisco_data}, the number of
service providers exploiting tiered pricing is increasing (e.g., the data plans of  {\em AT$\&$T} ). We consider a
tiered pricing model that gives two different limitations for the
amount of mobile data service from
macro BS with different prices. Thus, for macro service, users should
select one of the two data plans. Additionally, users can use unlimited
data from femto BSs when they subscribe femto services.


 At first, the provider makes two data plans to maximize the
following problem:
\begin{eqnarray}
\label{eq:tiered}
{\bf Provider:} &&   \max_{p^{  Plus },p^{  Pro },p_{o},
  p_{c},x^{Plus},x^{Pro} \ge 0} R, 
\end{eqnarray}
where $x^{Plus}$ and $x^{Pro}$ are data limit for data plan $Plus$ and
$Pro$, respectively. Note that, when the provider makes data plans, both price
and data limit should be decided. In tiered pricing scheme, the revenue
is simply calculated as follow:  
\begin{equation}
 \label{eq:revenuetiered}
  R=N \Big \{ \sum_{t \in \{   Plus, Pro  \} } p^{t} \alpha^{t}
 + \sum_{j \in  \{o,c \} } ( p_{j} -c_f ) \cdot \alpha_{j} \Big \},
\end{equation}
where $\alpha^{Plus}$ and $\alpha^{Pro}$ are subscription ratios for
each data plan. 

Users want to maximize their utilities. In this pricing scheme,
there are two options for macro BS's data plan. Moreover, users can
select one of service types among  {\em mobile-only}, {\em mobile+open
  femto}, and {\em mobile+closed femto}. Thus, the solution of
following equation becomes the choice of each user.
\begin{eqnarray}
{\bf User:} && j^*(\gamma), t^*(\gamma) =\arg \max_{  j ,t}  \tilde{U}_{j,t}({\bm{x}};\gamma),
\end{eqnarray}
subject to
$$ \alpha^{Plus} x^{Plus} + \alpha^{Pro} x^{Pro} \le C_M , $$
where $j \in \{ o,c \}$, $t \in \{Plus, Pro \}$.

\subsection{Equilibrium}

Finding equilibrium in tiered pricing is similar to the procedure in
flat pricing. The prices ($p^{Plus}$, $p^{Pro}$,
$p_{o}$, and $p_{c}$) are functions of subscription ratio $\alpha$ values ($\alpha^{Plus}$, $\alpha^{Pro}$,
$\alpha_{o}$, and $\alpha_{c}$) and data limits($x^{Plus}$ and
$x^{Pro}$). Thus, when we fix the $\alpha $ values,  the revenue of provider
becomes a function of data limits. Therefore, for fixed $\alpha$ values, the
revenue maximization problem is described as follows:
\begin{eqnarray}
\label{eq:tieredx}
\max_{x^{Plus},x^{Pro} \ge 0}&&   R(x^{Plus},x^{Pro} ), \\
s.t. &&  \alpha^{Plus} x^{Plus} + \alpha^{Pro} x^{Pro} \le C_M.
\end{eqnarray}

Let $x^{Plus}({\bm \alpha})$ and $ x^{Pro}({\bm \alpha})$ be the
solution of above problem. In this subsection, let the ${\bm \alpha}$
denote the vector $(\alpha^{Plus},\alpha^{Pro},\alpha_{o},\alpha_{c})$. Then, the revenue of provider can be
represented as the function of subscription ratios. So, the
equilibrium point is derived from the following maximization problem.
\begin{eqnarray}
\label{eq:tieredalpha}
\max_{{\bm \alpha} }&&   R({\bm \alpha}  ), \\
\label{eq:condition} s.t. && \alpha_{o}+\alpha_{c} \le \alpha^{Plus}+\alpha^{Pro}, \\
      && {\bm 0} \le {\bm \alpha},
\end{eqnarray}
Note that, the
number of users, who subscribe femto services, should be less than the
number of entire subscribing users as the condition~(\ref{eq:condition}).
Recall that, when
the ${\bm \alpha}$ of equilibrium point is found, we can get
$x^{Plus}$ and $x^{Pro}$ for the ${\bm \alpha}$ from the solution of
the eq.~(\ref{eq:tieredx}). Moreover, the prices are determined by the
subscription ratios and data limits.


%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "main"
%%% End: 